论文标题
$ 3 $ - 发育的椭圆曲线的Selmer组$ y^2 = x^3+n^2 $
The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$
论文作者
论文摘要
考虑椭圆曲线的家族$ e_n：y^2 = x^3+n^2 $，其中$ n $在阳性cubefree整数中变化。从$ e_n $到$ \ hat {e} _n：y^2 = x^3-27n^2 $，有一个合理的$ 3 $ - 发育$ ϕ $，以及双重差异$ \ hat ϕ：\ hat {e} _n \ hat {e} _n \ rightarrow e_n $。我们表明，几乎所有$ n $，$ \ mathrm {sel} _ϕ（e_n）$的排名是$ 0 $，$ \ mathrm {sel} _ {\ hat \ tartical}（\ hat \ hat {e} _n）$的等级是由$ n $ ye ye ye y ye y y ye y y ye y y y ye BMM的数量确定9 $。
Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $ϕ$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hatϕ:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_ϕ(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hatϕ}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.
